"If the expenses for AB and the common house expenses are really 50-50 ..."
You're still lumping together expenses of the home and support for the child. Head of household depends only on the cost of keeping up the home, not support of any child. Reread my earlier post above about the cost of keeping up the home.
"... then can the parties just make an agreement between them on which will use HOH on their return?"
No, as I said earlier, only the one who paid more than half the cost of keeping up the home can file as head of household.
"Can you file as single but also list dependents without being an HOH?"
Yes.
